How do you calculate this limit?

[itsrayex]

I have tried to solved it like three times. Its supposed to yeld infinite but I keep getting undefined form. What is the correct way to solve this?

 

[JeffM]

What does your text list as indeterminate forms?

[math]x > e \implies x^2 + 2 > e \implies ln(x^2 + 2) > 1.\\ \therefore x > e > 0 \implies x * ln(x^2 + 2) > x * 1 = x.[/math]
Now apply the formal definitions of the limit as x approaches infinity and the limit equals infinity.

I point out that [imath]log(x^2 + 2) \ne log(x^2) + log(2) = log(2x^2).[/imath]

 

[BigBeachBanana]

[math]\lim_{x\to \infty}x\log(x^2+2)=(\lim_{x\to \infty}x)(\lim_{x\to \infty}\log(x^2+2))=\infty\cdot\infty=\infty[/math]

 

[itsrayex]

What does your text list as indeterminate forms?

[math]x > e \implies x^2 + 2 > e \implies ln(x^2 + 2) > 1.\\ \therefore x > e > 0 \implies x * ln(x^2 + 2) > x * 1 = x.[/math]
Now apply the formal definitions of the limit as x approaches infinity and the limit equals infinity.

I point out that [imath]log(x^2 + 2) \ne log(x^2) + log(2) = log(2x^2).[/imath]

Turns out I mistakenly thought infinite*infinite is an indeterminate form, but it's not. Thank you!